1 atmospheremodel (cam)

JAMES, VOL. ???, XXXX, DOI:10.1002/,

Energy considerations in the Community1

Atmosphere Model (CAM)2

David L. Williamson,1Jerry G. Olson,

1Cecile Hannay,

1Thomas Toniazzo,

2

Mark Taylor,3Valery Yudin,

1,4

Corresponding author: David L. Williamson, National Center for Atmospheric Research, Box

3000, Boulder, CO 80307-3000, USA. ([email protected])

1National Center for Atmospheric

Research, Boulder, Colorado

2Uni Research and Bjerknes Centre for

Climate Research, Bergen, Norway

3Sandia National Laboratories,

Albuquerque, New Mexico

4University of Colorado, CIRES, Space

Weather Prediction Center, Boulder,

Colorado

D R A F T May 19, 2015, 2:31pm D R A F T

X - 2 WILLIAMSON ET AL.: ENERGY CONSIDERATIONS IN CAM

Abstract. An error in the energy formulation in the Community Atmo-3

sphere Model (CAM) is identified and corrected. Ten-year AMIP simulations4

are compared using the correct and incorrect energy formulations. Statistics5

of selected primary variables all indicate physically insignificant differences6

between the simulations, comparable to differences with simulations initial-7

ized with rounding sized perturbations. The two simulations are so similar8

mainly because of an inconsistency in the application of the incorrect energy9

formulation in the original CAM. CAM used the erroneous energy form to10

determine the states passed between the parameterizations, but used a form11

related to the correct formulation for the state passed from the parameter-12

izations to the dynamical core. If the incorrect form is also used to deter-13

mine the state passed to the dynamical core the simulations are significantly14

different. In addition, CAM uses the incorrect form for the global energy fixer,15

but that seems to be less important. The difference of the magnitude of the16

fixers using the correct and incorrect energy definitions is very small.17


WILLIAMSON ET AL.: ENERGY CONSIDERATIONS IN CAM X - 3

1. Introduction

Atmospheric models represent highly complex, nonlinear processes which continually18

interact with each other in space and time. Some components such as the atmospheric19

flow are of relatively large scale and can be approximated with a variety of numerical20

methods common in computational fluid dynamics. Other components, such as precipi-21

tation processes, occur on small scales, often finer than the scales used to represent the22

fluid flow, and require other approaches to approximate them.23

Because of this difference in scales, atmospheric models are conceptually divided into24

two primary components commonly referred to as the dynamical core and the param-25

eterization suite. The dynamical core approximates the resolved fluid (air) flow of the26

three-dimensional atmosphere. The discrete representation of the dynamical core gen-27

erally defines the grid points and/or grid cells underlying the approximations. In order28

for the model to be computationally tractable the areas associated with those points and29

cells are generally larger than the scale of the physical processes important for climate.30

Therefore those processes must be parameterized in the model. The parameterization31

suite attempts to approximate this subgrid-scale forcing in terms of grid-scale properties32

from the dynamical core, hence the term parameterization.33

The parameterization component itself consists of many interrelated and interacting34

complex nonlinear processes and is thus further divided into sub-components in order to35

make the collection practical to solve, hence the reference to a suite of processes. The36

processes considered individually typically include deep convection, shallow convection,37

surface exchange, planetary boundary layer turbulent mixing, longwave and shortwave38



radiation, cloud formation and evolution. Since the processes are subgrid-scale and depend39

on the grid box averages, the approximations for each process are formulated for a single40

horizontal grid box, independent of its neighbors. However, the approximations generally41

involve the vertical column of grid boxes through the depth of the atmosphere. Each42

column is solved independently of its neighboring columns.43

Beginning with the first version of the Community Atmosphere Model, labeled CAM2,44

the individual components in the parameterization suite in the CAM series have calcu-45

lated and applied the tendencies in a time-split manner [Collins et al., 2003]. In that46

splitting, each parameterization component updates the state; the ensuing parameteriza-47

tion component starts from the state updated by the preceding component, and in turn48

updates the state further. CAM has several dynamical cores available that combine dif-49

ferently with the parameterization suite. The finite volume dynamical core, considered50

here, is coupled to the parameterization suite in a time-split manner, and receives as input51

the updated state resulting from the last component of the parameterization suite. By52

contrast, the spectral transform Eulerian and semi-Lagrangian cores are coupled to the53

parameterization suite in a process-split manner in which both components start from54

the same state. The result of the parameterization suite is applied as a forcing in the55

dynamical core. The terminology used to designate different splitting methods is not56

universal, see Williamson [2002] for more complete descriptions of such terminology used57

in association with CAM. In this paper, we limit our discussion to the time-split form of58

the finite volume dynamical core.59

The conservation of total energy (including internal, kinetic, and potential energy) is a60

property of the continuous atmospheric equations and should also be a property of the dis-61



crete, time-split numerical approximations. Energy conservation could be easily achieved62

if total energy were made a prognostic variable and prognosed with conservative numerical63

schemes. However, that does not guarantee an accurate solution and total energy is not64

a prognostic variable in almost all atmospheric model formulations. Nevertheless, energy65

must be conserved to a minimal level in atmospheric models when they are coupled to66

ocean, sea-ice and land models intended for long climate simulations. Boville [2000] origi-67

nally suggested the atmospheric component should conserve energy to at least 0.1 W m−268

to avoid spurious long-term trends in the coupled system. However, for centuries-long69

climate projections it is probably safer to conserve to 0.01 W m−2. Such conservation can70

be obtained with the application of an energy fixer as discussed below.71

In the parameterization suite each process is formulated and solved individually. Thus,72

each processe should conserve energy individually in the sense that the energy change by73

the process equals the net source/sink calculated by that process. When the dynamical74

core is time-split from the parameterization suite the core provides an approximate solu-75

tion to the source-free continuous fluid equations. In energy terms, the processes in the76

dynamical core include transport of energy and conversion of potential to kinetic energy,77

under conservation of the global integral of total energy. In addition, kinetic energy dis-78

sipation either from viscous processes represented explicitly as a term in the momentum79

equations, or implicitly as a property of the numerical approximations, should conserve80

energy by contributing heat to the fluid. A heating associated with explicit viscous terms81

can often be derived and included in the approximations as is done in the CAM spectral82

transform dynamical core [Collins et al., 2004; Neale et al., 2010a, b] and the spectral83

element dynamical core [Taylor , 2011; Neale et al., 2010b]. However, such heating might84



not truly represent the physics of the frictional energy transformation. Viscous terms85

introduced as horizontal diffusion could be treated as a separate parameterization. How-86

ever they are generally considered as part of the dynamical core, in part because they87

involve horizontal neighbors and are often implemented for pragmatic reasons to control88

numerical noise, and in part because the numerical approximations may generate addi-89

tional damping as a numerical artifact [Jablonowski and Williamson, 2011]. Such implicit90

damping is difficult to determine locally but the global average value can be obtained as91

a residual. In such cases a global energy fixer can be applied. For example the semi-92

Lagrangian dynamical core version of CAM3 uses a form described in Williamson et al.93

[2009] and Jablonowski and Williamson [2011] while the finite volume dynamical core in94

CAM uses a different form discussed in Neale et al. [2010a, b]. These fixers add a uniform95

increment to the temperature field to compensate for the global average energy lost by the96

dynamical core that time-step. While this ensures a global energy balance, any impact of97

the conservation error would be in the spatial distribution which cannot be determined.98

In the time-split approach, the subgrid-scale parameterizations need to calculate changes99

in the energy associated with sources and sinks. Since the parameterizations are formu-100

lated for a grid column, the integral of the energy in the column at the end of the process101

should equal the integral at the beginning of the process plus the net source given by the102

fluxes through the column. Boville and Bretherton [2003] derive the form of energy to be103

conserved within the parameterization suite and present a method to update the atmo-104

spheric state so that their energy is conserved at all stages within the parameterization105

suite. Their form of energy is also used in CAM for the global energy fixer associated106

with the finite volume dynamical core. Unfortunately, the energy they derive is not the107



appropriate form for the system of equations used in CAM. In the following we summa-108

rize their development, explain why their form is inappropriate, describe the necessary109

corrections to the model formulation, and discuss the impacts on the model simulations.110

2. Energy Equations

Boville and Bretherton [2003] derive a total energy equation in the height coordinate111

system with the goal of constructing energy conservative parameterizations in CAM. That112

equation, their Eqn. (9), slightly simplified here with regard to the notation for the fluxes,113

takes the form114

d

dt(K + cpT + Φ) =

1

ρ

∂p

∂t+ Fnet (1)115

where K ≡ v · v/2, v is the vector velocity, T is temperature, ρ is density, p is pressure,116

t is time and cp is specific heat capacity of dry air at constant pressure. The term Fnet117

here includes the last two terms in Boville and Bretherton Eqn. (9). The geopotential,118

Φ, is related to the temperature by the hydrostatic equation. The net fluxes calculated119

by the parameterizations, i.e. the heating and momentum forcing, are denoted Fnet.120

Here we follow Boville and Bretherton [2003] and do not include the energy associated121

with water in its various forms which could be included in the conservation equation,122

so Fnet also includes heating/cooling associated with the phase changes of water. Water123

is assumed to be conserved by the numerical approximations. In CAM, the individual124

parameterizations do not change pressure, and do not include dynamical processes such125

as resolved advection since those are handled by the time-split dynamical core. Thus for126

application to the parameterizations in a column Boville and Bretherton [2003] simplified127



Eqn. (1) to128

∂

∂t(K + cpT + Φ) = Fnet (2)129

The implementation of Eqn. (2) in CAM adopted a simple forward differencing which for130

temperature updated by the ith parameterization component in the time-split sequence131

can be written132

cpTi + Φ

(

T i)

= cpTi−1 + Φ

(

T i−1)

+∆tF (T )inet (3)133

where T i−1 is the state from the previous component, F (T )inet is the thermal energy ten-134

dency from the ith component and T i is the updated state. Since Φ(T ) depends on T , the135

combination [cpTi + Φ(T i)] can be inverted to obtain T i. Boville and Bretherton [2003]136

describe how this is done in CAM. Similar update equations are applied in CAM for137

momentum and thus the kinetic energy component. These terms are treated correctly in138

Boville and Bretherton [2003] and thus we do not include them here. Only the thermo-139

dynamic component needs correction. Eqns. (1) and (2) were derived for the z vertical140

coordinate but applied to CAM which is based on transformed pressure vertical coordi-141

nates. Those equations do not apply in that system. We derive the corresponding form142

for the hybrid-pressure vertical coordinate of CAM shortly.143

CAM also incorrectly implemented a global energy fixer based on the energy defined144

in Eqn. (2). The fixer conserves the vertical and global integral of that form since the145

dynamical core calculates energy exchanges along with transport which are not necessarily146

local. As is the case in CAM, dynamical core numerical approximations are often derived147

to conserve the average of the conversion of potential energy to kinetic energy. In such148

models the global energy fixer is intended to compensate for energy loss from inherent149



numerical dissipation, and non-conservation due to time truncation errors. It may also150

include other non-conservative numerical processes such as vertical remapping or possibly151

errors in the parameterizations.152

We now summarize the global energy integrals appropriate for conservation by the dy-153

namical core in CAM and then derive the local form appropriate for the parameterization154

updates following the approach of Boville and Bretherton [2003]. Laprise and Girard155

[1990], following Kasahara [1974], derive the appropriate equations in the hydrostatic156

transformed pressure coordinates:157

∂

∂t

∫

A

ηs∫

ηtop

(K + cpT )∂p

∂ηdη + psΦs

dA =

∫

A

ηs∫

ηtop

Fnet

∂p

∂ηdη dA (4)158

An equivalent form is159

∂

∂t

∫

A

ηs∫

ηtop

(K + cvT + Φ)∂p

∂ηdη + ptopΦtop

dA =

∫

A

ηs∫

ηtop

Fnet

∂p

∂ηdη dA (5)160

with cv denoting the specific heat at constant volume [Neale et al., 2010b, Section 3.2.2].161

The transform pressure vertical coordinate is denoted by η, subscripts s and top denote162

the bottom (surface) and top of the model, respectively, and the integral dA denotes the163

global horizontal integral. It is immediately apparent that the energy form in Eqn. (2)164

is inconsistent with either form appropriate for the dynamics, Eqn. (4) or Eqn. (5). The165

dynamics equation involving cp, Eqn. (4), does not include Φ in the vertical integral and166

the equation which includes Φ in the vertical integral, Eqn. (5), has cv instead of cp.167

When the dynamical core is time-split from the parameterization components as with168

the finite volume core there is no net forcing and the right-hand side Eqn. (4) or (5)169

should be zero in CAM. As explained in Boville and Bretherton [2003], generally, if the170

model includes an explicit horizontal momentum diffusion to stabilize the numerical ap-171



proximations or to shape the tail of the energy spectrum, a compensating heating can be172

added to give zero net forcing. However, if the numerics contain inherent damping, or if173

other diffusion terms are added to the dynamics, a global “energy fixer” is generally added174

to yield energy conservation since the associated local damping is difficult or impossible to175

determine and compensate [Jablonowski and Williamson, 2011]. CAM-FV has inherent176

numerical damping and thus applies a global energy fixer to obtain conservation [Neale177

et al., 2010b]. However, rather than being based on Eqn. (4) or Eqn. (5) that fixer is178

based on the global integral of the form of energy in Eqn. (2). The assumptions that went179

into Eqn. (2) are clearly inappropriate for the dynamical core. We do not know why this180

energy was chosen, unless it was thought to be more consistent with the parameteriza-181

tions, or perhaps a stable climate with a small global average net energy flux could not be182

obtained from the parameterizations in a long simulation when the dynamical core and183

parameterizations conserved different energies.184

We now derive the local energy equation for the hydrostatic transformed pressure coor-185

dinates of CAM following the approach of Boville and Bretherton [2003]. Starting with the186

thermodynamic equation in transformed pressure coordinates, Laprise and Girard [1990]187

Eqn. (2.2), adding dΦ/dt to both sides and substituting the hydrostatic equation gives188

d

dt(cpT + Φ) =

∂Φ

∂t+

RT

p

∂p

∂t+ cpQ+ v · ∇Φ +

RT

pv · ∇p (6)189

where R is the gas constant for moist air and Q is the parameterized sub-grid scale190

heating. Starting with the momentum equation in transformed pressure coordinates,191

Laprise and Girard [1990], Eqn. (2.1), and taking the dot product with v gives, after192



some manipulation,193

d

dt(K) = −v · ∇Φ− v · (RT∇ ln p) + v · F (7)194

where F is the parameterized momentum forcing. Adding Eqn. (6) and Eqn. (7) gives195

d

dt(K + cpT + Φ) =

∂Φ

∂t+

1

ρ

∂p

∂t+ Fnet (8)196

which has an additional term compared to the Boville and Bretherton [2003] form, Eqn.197

(1). For the CAM parameterizations, where pressure is not changed and dynamics is198

absent, Eqn. (8) simplifies to199

∂

∂t(K + cpT ) = Fnet (9)200

Eqn. (9) is completely consistent with Eqn. (4). The parameterizations can be updated201

by202

cpTi = cpT

i−1 +∆tF (T )inet (10)203

rather than Eqn. (3) and the dynamical core global energy fixer can be based on Eqn.204

(4) with complete consistency.205

We note that with the application of time-splitting each parameterization that changes206

water vapor should change the pressure because pressure in CAM is defined to be moist.207

However, the individual parameterizations in CAM do not change the pressure. Instead,208

after the entire parameterization suite is completed, the pressure is corrected in each209

layer to account for the net water vapor change which preserves the dry mass of the atmo-210

sphere. At the same time, constituent specific ratios are modified to conserve constituent211

masses. The moisture-related change in pressure also has energy implications associated212



with energy of the non-vapor water components. Boville and Bretherton [2003] (end of213

Section 3) state that a form conserving the energy transferred to and from the non-vapor214

components was being tested but apparently it was not successful and was not adopted215

in the model. This moisture effect energy conservation discrepancy, about 0.3 W m−2216

global-annual average sink in CAM, was folded into the global energy fixer associated217

with the dynamical core. We do not discuss this further here, but work to rectify this218

issue is underway.219

3. Simulations

We have implemented the correct energy in the parameterization updates and in the220

global energy fixer associated with the finite volume dynamical core in CAM5.2 and carried221

out a 1 degree AMIP type simulation starting from 1 January 1979. CAM5.2 is the atmo-222

spheric component of CESM1.1 (see http://www.cesm.ucar.edu/models/cesm1.1/cam.)223

In all simulations presented here all free parameters are set to the standard CAM5.2 val-224

ues. We present 10-year annual averages of a few variables from the simulation averaged225

for 1980 to 1989. These are compared to a matching control simulation with the standard226

1 degree CAM5.2. In the following these simulations are labeled CORRECT and CAM,227

respectively. These and other experiments are summarized in Table 1. The distinction228

between the two columns giving T passed to the parameterizations and T passed to the229

dynamical core will become clear after Eqn. (11) is introduced. The code flow for the230

simulations is also summarized in Fig. 1.231

Table 2 compares ten-year annual average, global averages of a few primary variables232

that are routinely examined when tuning the model. These are a subset of the many233

considered during model development. The averages from the two simulations (columns234



one and three) are remarkably close. In fact one might think these are just from two235

different realizations of the same model rather than from two different models.236

To address this possibility we ran a simulation with the corrected model starting with237

a perturbed initial condition - a rounding sized random increment was added to the238

temperature in the initial file. The second column of Table 2 labeled CORRECT/PERT239

gives the global averages for this simulation. The differences between CORRECT and240

CAM are of similar magnitude to the differences between the runs with different initial241

conditions. None are physically significant.242

Figure 2 (top) shows the ten-year annual average, zonal average temperature difference243

between the simulations with the correct energy formulation (CORRECT) and with CAM.244

These differences are also remarkably small, being less than 0.25K over most of the domain.245

The maximum difference is just over 0.5K in the southern lower polar stratosphere. The246

middle panel shows the difference between the two simulations with the correct energy247

formulation (CORRECT and CORRECT/PERT). Recall the only difference is the initial248

condition. The differences are comparable in magnitude to those in the top panel but the249

structures are slightly different. Table 3 shows the RMS differences of ten-year annual250

averages of selected horizontal fields. The left column contains CORRECT minus CAM.251

The middle contains CORRECT minus CORRECT/PERT. As with the other measures252

the RMS differences between the two different models are very small and comparable to253

the differences from the perturbation simulation.254

One might wonder why the differences associated with the different energy definitions255

are so small. CAM uses Eqn. (3) to update the temperature after each parameteriza-256

tion and passes that temperature to the next parameterization, while CORRECT uses257



Eqn. (10). However, after the last parameterization in CAM the final temperature from258

Eqn. (3) is not passed to the dynamical core. Instead, unexpectedly, a final tempera-259

ture is passed that is calculated from the sequence of fluxes F (T )inet determined by the260

parameterizations.261

T I = T 0 +1

cp

I∑

i=1

∆tF (T )inet (11)262

At first glance this looks consistent with updating the temperature using the correct energy263

Eqn. (10). However the temperature T i−1 which was input to the parameterization to264

calculate F (T )inet comes from Eqn. (3) rather than from Eqn. (10) and the two input265

temperatures are only the same for the first parameterization called in the suite. Boville266

and Bretherton [2003] do describe this calculation of the final temperature in the top267

left column of page 3884 stating that this leads “to a small energy imbalance” that “will268

be addressed in a future model revision.” We have calculated the time average, global269

average of this energy imbalance in CAM5.2 (i.e in terms of the Boville and Bretherton270

[2003] energy) to be effectively a source of 0.9 W m−2, which is absorbed into the global271

average energy fixer applied after the dynamics. We do not know why this choice was made272

in CAM. In fact a code comment refers to it as a “kludge”. We queried C. Bretherton and273

he replied that turbulent dissipation heating due to momentum diffusion was his main274

contribution to the paper and he was not sure why Boville ultimately introduced that275

kludge (personal communication, 2013). At the end of this section we will show the effect276

of passing T from Eqn. (3) instead of from Eqn. (11) to the dynamical core.277

It is also not clear why using the temperature from Eqn. (3) in the parameterizations278

in CAM instead of that from Eqn. (10) as in the corrected model seems to have such279

a small effect on the heating rates calculated by the parameterizations. Both CAM and280



CORRECT essentially use Eqn. (11) to obtain the temperature passed to the dynamical281

core, the only difference being the sequence of temperatures passed between the parame-282

terizations and thus defining the input values to the parameterizations. We might expect283

more of an accumulated effect in fluxes calculated by the parameterizations themselves,284

but this appears not to be the case. It is possible that there is a compensation between285

the different processes, in which a change in heating by one process is offset by an oppo-286

site change in a following process, especially with the time-split formulation. This does287

not appear to be the situation here. We have examined the differences between the two288

experiments in the heating from individual processes. There is only a small compensation289

between the shallow convection and the macrophysics. The difference in the total heating290

does seem to be an accumulation over the processes with little compensation.291

Another difference between the two models is the energy definitions used in the global292

average energy fixers. It is possible that this difference offsets differences in the param-293

eterized fluxes. To examine this possibility we did an additional simulation modifying294

CAM to pass T from Eqn. (10), i.e. the correct formulation, to the parameterizations295

rather than T from Eqn. (3), the incorrect formulation, but continuing to use the incorrect296

energy formulation for the global energy fixer. This simulation is labeled CAM/PARAMS297

CORRECT. The resulting ten-year annual averages are shown in Fig. 2 and the Tables.298

Table 2 presents the global averages from this simulation in the last column. They are299

very close to the other simulations. However, the net energy fluxes are closer to the two300

simulations with the correct energy formulation than to CAM, presumably reflecting the301

different states passed to the parameterizations. This implies that the energy formulation302

used for the global energy fixer has less effect. In fact, the difference between the fixers303



from the cases CORRECT and CAM is very small, the 10-year average difference being304

1.52× 10−5 K/day compared to 4.30× 10−3 K/day and 4.32× 10−3 K/day for the values305

themselves for CORRECT and CAM simulations, respectively. The right column of Table306

3 shows the RMS differences of CAM/PARAMS CORRECT with CAM. The differences307

are similar to the others in the table, perhaps slightly larger for a few variables but not308

physically significant. Figure 2 bottom shows the zonal average temperature difference309

between CAM/PARAMS CORRECT and CAM. The structure of the difference resem-310

bles that of the difference between CORRECT and CAM, but the amplitude is slightly311

larger. This also implies that the structure of the difference is likely due to the different312

fields passed between the parameterizations and thus the heating passed to the dynamics,313

rather than to the energy formulation applied in the global fixer. However the differences314

are still quite small, comparable to the differences from the perturbation run.315

In a single time step, after each parameterization the difference in T from Eqn. (3)316

(used by CAM) and from Eqn. (11) (but accumulated only through the previous pa-317

rameterizations and calculated as a diagnostic) is rather small. The top panel of Fig.318

3 shows the ten-year annual average zonal-mean of the difference of temperature after319

the last parameterization of the suite, calculated according to Eqns. (3) and (11) from320

the standard CAM simulation which used Eqn. (3) for the parameterization updates but321

passed the value from Eqn. (11) to the dynamical core. The largest average differences322

are 0.01K. The difference in fluxes calculated by the parameterizations are probably also323

relatively small. However we are not able to determine that without a major change to324

the model to allow a second, diagnostic calculation of each parameterization based on the325

other temperature. Apparently however, the difference in fluxes has little effect on the326



simulation as indicated by CAM/PARAMS CORRECT in the Tables and Figures. On327

the other hand, if the temperature from Eqn. (3) is passed to the dynamical core, i.e. if328

CAM had used the incorrect energy formulation consistently, the effect on the simulations329

becomes significant. This is seen in the global averages in Table 2 and in the bottom panel330

of Fig. 3 which shows the difference of a simulation with the standard CAM5.2 minus a331

simulation with CAM5.2 where the updates that use Eqn. (3) are not replaced by the332

kludge at the end of the parameterization suite. This latter simulation is labeled INCOR-333

RECT. Note the contour interval in Fig. 3 is 10 times larger than in Fig. 2 and that the334

ordinate is logarithmic rather than linear since the largest differences are at and above the335

tropopause where they reach maxima of about 10K near the poles. Typical errors in the336

tropics and in the mid-latitudes are of the order of 1-2K. The meridional dependence of337

the climate’s sensitivity to the parameterization updates is consistent with a stronger can-338

cellation between diabatic and dynamic heating tendencies characteristic of the tropics.339

The total diabatic heating rates for example show systematic differences of 10% between340

the simulations in the annual means. Seasonal means show larger differences still. Also341

noteworthy are systematic regional differences in the net total heat flux at the surface,342

which have implications for coupled simulations with an interactive ocean component and343

lead to systematically different simulated SST patterns. Nevertheless, given the difference344

in vertical structures arising from the different temperature calculations shown in the top345

panel of Fig. 3 and in Fig. 1 of Boville and Bretherton [2003] we might have expected346

larger differences in the tropics. However those differences interact with the dynamics to347

create the different climates.348



The time-split structure and energy conservation issues of CAM5.2 discussed349

above are not restricted to the finite volume dynamical core. The spectral ele-350

ment core shares the same structure. Since the correct energy formulation was351

proposed for inclusion in CAM5.4, as part of the development evaluation it was352

further tested in standalone simulations with both the finite volume and spec-353

tral element dynamical cores in CAM5.3. Such standalone simulations were car-354

ried out for most of the candidate changes. These simulations are documented at355

www.cesm.ucar.edu/working groups/Atmosphere/development/cam6/cam5.4/. Atmo-356

spheric Model Working Group (AMWG) standard diagnostics comparing the simulations357

from CAM5.3 modified to use the correct energy with ones from standard CAM5.3, which358

continued to use the incorrect energy formulations of CAM5.2, are reachable from that359

site under categories C8 and C8b for the finite volume and spectral element cores, re-360

spectively. Although the standard contour intervals used there are not as discriminating361

as used in this paper, there is no indication that the conclusions drawn here with the362

finite volume core are invalid for the spectral element core. On that web site, the differ-363

ences introduced by the energy definition changes can also be compared with differences364

introduced by other changes proposed during the CAM5.4 development.365

4. Summary

An error in the energy formula used in CAM is identified. The error has percolated366

through all versions of CAM up to and including CAM5.2. The incorrect form of energy367

was derived and used to conserve energy when updating the time-split components within368

the parameterization suite. It was originally derived for non-hydrostatic and hydrostatic369

height coordinates but applied to hydrostatic hybrid pressure coordinates. We derive the370



correct form of energy for application to the parameterization suite for the hydrostatic371

hybrid pressure system. The incorrect form was also used in the global energy fixer applied372

with the finite volume dynamical core, but not in the fixer applied to the other dynamical373

cores available in CAM.374

We implemented the correct energy in the parameterizations and in the global energy375

fixer and carried out a long simulation. We present 10-year annual averages of AMIP376

simulations from the corrected model and from the original model. We present a few377

global averages which indicate insignificant changes in cloud radiative properties, in the378

net energy fluxes at the top and bottom of the atmosphere and in the precipitation and379

precipitable water. The changes are comparable to natural variability determined by380

a second simulation with the correct energy formulation but starting from a different381

initial condition. The zonal average temperature differences are also insignificant, as382

are RMS differences for selected horizontal fields. The primary reason the differences383

are not significant is that the incorrect energy was not used consistently in the original384

CAM. It was used for the global energy fixer and to determine the state passed between385

parameterizations. However, the final temperature from the parameterization suite that386

was passed to the dynamical core was calculated from the parameterized fluxes applied387

in a manner consistent with the correct energy. On the other hand, when the incorrect388

energy is used consistently, i.e. when the state passed to the dynamical core from the389

parameterized fluxes is determined using the incorrect energy, the simulation is affected390

significantly. In this case all aspects of the model are based on the incorrect energy.391

The major differences are around and above the tropopause. Application of the incorrect392

energy for the global energy fixer has an insignificant effect. The difference between the393



fixers using the different energy definitions was 0.05% of the fixers themselves. The results394

here are based on AMIP simulations with specified sea-surface temperatures. There might395

be small, local systematic differences in surface fluxes that affect coupled simulations.396

However, in developing CAM5.4 any such effect on coupled runs has been small compared397

to changes in the parameterizations.398

Acknowledgments. We would like to thank J. J. Tribbia for comments during the399

course of this work, Peter Caldwell for suggestions which greatly improved the original400

manuscript, and an anonymous reviewer for suggesting the inclusion of Fig. 1. The401

data used in this paper may be obtained by directing requests to any of the first three402

NCAR authors (Williamson, [email protected]; Olson, [email protected], or Hannay, han-403

[email protected].) DLW and JGO were partially supported by the Regional and Global Cli-404

mate Modeling Program (RGCM) of the U.S. Department of Energy’s, Office of Science405

(BER), Cooperative Agreement DE-FC02-97ER62402. The National Center for Atmo-406

spheric Research is sponsored by the National Science Foundation.407

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Table 1. Simulation summary with equations used by different components

LABEL DESCRIPTION T passed T passed Energyto paramsa to dynamb Fixerc

CORRECT CAM5.2 with Correct (10) Correct (10) Correct (4)correct energy

CAM CAM5.2 CONTROL Incorrect (3) Correct (11) Incorrect (2)simulation

CORRECT/ initial perturbation added Correct (10) Correct (10) Correct (4)PERT to CORRECTCAM/PARAMS CAM5.2 modified to pass Correct (10) Correct (11)d Incorrect (2)CORRECT correct energy to

parameterizationINCORRECT CAM5.2 modified to pass Incorrect (3) Incorrect (3) Incorrect (2)

incorrect but consistentenergy to dynamics

a Equation for T passed to the next parameterization.

b Equation for T passed to the dynamical core.

c Equation for energy used in global energy fixer with dynamical core.

d When Eqn. (10) is used for T passed to the parameterizations, Eqn. (11) is equivalent to

Eqn. (10) for T passed to dynamical core.

Table 2. Ten-year annual average, global averages

VARIABLE CORRECT CORRECT/ CAM CAM/PARAMS INCORRECTPERT CORRECT

Net energy flux (W m−2)a

Top of model 0.485 0.479 0.558 0.485 3.866Surface 0.477 0.480 0.542 0.458 3.833

Cloud fraction (%)High 37.590 37.541 37.545 37.623 40.562Low 41.936 41.984 41.894 41.982 42.322Middle 25.700 25.673 25.713 25.671 25.679Total 63.144 63.173 63.109 63.196 65.348

Cloud forcing (W m−2)Longwave 22.395 22.393 22.447 22.425 25.017Shortwave −48.677 −48.665 −48.677 −48.720 −49.548

Precipitation (mm day−1) 3.029 3.028 3.029 3.027 2.921Precipitable water (mm) 25.125 25.097 25.124 25.125 25.003

a Positive downward.



Table 3. RMS differences of ten-year annual averages

CORRECT CORRECT CAM/PARAMSversus versus CORRECT

VARIABLE CAM CORRECT/ versusPERT CAM

Surface Pressure (mb) 0.42 0.43 0.48200 mb Temperature (K) 0.17 0.17 0.21850 mb Temperature (K) 0.18 0.18 0.18200 mb zonal wind (m s−1) 0.65 0.69 0.83500 mb Geopotential height (m) 0.05 0.05 0.06Precipitation (mm day−1) 0.19 0.02 0.19Precipitable water (mm) 0.29 0.29 0.30Longwave cloud forcing (W m−2) 0.96 0.99 0.94Shortwave cloud forcing (W m−2) 1.65 1.64 1.68



CORRECT CAM

T i−1

PROCESS i

F (T )inet❄✤

✣✜✢

ENERGY CONSERVINGUPDATE EQN. 10

T i

❄✛

✻

i = 1, ..., I

✻✲

❄

❄TI

✛

✛✚

✘✙

PASS STATET I

✛

✻

T phys

DYNAMICALCORE

✻

T dyn

✛✚

✘✙

ENERGY FIXEREQN. 4

✻

✻✲

T 0❄

T i−1

PROCESS i

F (T )inet❄✤

✣✜✢


T i

❄✛

✻

i = 1, ..., I

✻✲

❄

❄∑I

1 F (T i)net✛

✛✚

✘✙

STATE UPDATEEQN. 11

✛

✻

T phys

DYNAMICALCORE

✻

T dyn

✛✚

✘✙

ENERGY FIXEREQN. 2

✻

✻✲

T 0❄

CAM/PARAMS CORRECT INCORRECT

T i−1

PROCESS i

F (T )inet❄✤

✣✜✢


T i

❄✛

✻

i = 1, ..., I

✻✲

❄

❄∑I

1 F (T i)net✛

✛✚

✘✙

STATE UPDATEEQN. 11

✛

✻

T phys

DYNAMICALCORE

✻

T dyn

✛✚

✘✙

ENERGY FIXEREQN. 2

✛✚

✘✙

✻

✻✲

T 0❄

GREEN = CORRECT

T i−1

PROCESS i

F (T )inet❄✤

✣✜✢


T i

❄✛

✻

i = 1, ..., I

✻✲

❄

❄TI

✛

✛✚

✘✙

PASS STATET I

✛

✻

T phys

DYNAMICALCORE

✻

T dyn

✛✚

✘✙

ENERGY FIXEREQN. 2

✻

✻✲

T 0❄

RED = INCORRECT

Figure 1. Schematic code flow diagrams illustrating the processes discussed in this paper for

the cases listed in Table 1. Green denotes correct energy used, red denotes incorrect used.



CORRECT − CAM

CORRECT − CORRECT/PERT

CAM/PARAMS CORRECT − CAM

Figure 2. Ten-year annual average, zonal average temperature differences. Top: simulation

with correct energy formulation minus simulation with standard CAM5.2. Middle: simulation

with correct energy formulation minus simulation with initial perturbation added to same model.

Bottom: simulation with CAM/PARAMS CORRECT, which passes T from Eqn. (10) between

parameterizations, minus simulation with CAM5.2, which passes T from Eqn. (3) between

parameterizations. Contour interval: 0.125 K.



DIFFERENCE IN CAMT FROM (11) MINUS T FROM (3)

CAM − INCORRECT

Figure 3. Top: Difference in temperature at the end of the parameterization suite obtained

from Eqn. (11) minus that obtained from Eqn. (3) in a single simulation with CAM5.2 which

uses the values from Eqn. (3) in the parameterizations. Values from Eqn. (11) were passed to the

dynamical core. Contour interval: 0.001K. Bottom: Difference of temperatures in two simulations

passing different temperatures from parameterization to dynamical core: CAM which passes T

from Eqn. (11) minus INCORRECT which passes T from Eqn. (3). Contour interval: 1 K.


1 atmospheremodel (cam)

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